where \(\operatorname{sym}(\cdot )\)  returns the symmetrical part of its argument.
The Riemannian gradient is sufficient in order to define a descent direction of \(f\) , i.e., a tangent vector inducing a decrease of \(f\) , hence to define a proper Riemannian steepest descent algorithm.
However, if one wants to resort to a more sophisticated optimization algorithm such as conjugate gradient or BFGS, vector transport  [1]} is required to transport a tangent vector from one tangent space to another.
In the case of \(\mathcal {S}^{++}_p\) , we can employ the most natural one, i.e., the one corresponding to the parallel transport associated with the affine-invariant metric [2]}.
The transport of the tangent vector \({\xi }\)  of \({\Sigma }\)  onto the tangent space of \({\hspace{0.83328pt}\overline{\hspace{-0.83328pt}\Sigma \hspace{-0.83328pt}}\hspace{0.83328pt}}\)  is given by
\(\mathcal {T}^{\mathcal {S}^{++}_p}_{{\Sigma }\rightarrow {\hspace{0.83328pt}\overline{\hspace{-0.83328pt}\Sigma \hspace{-0.83328pt}}\hspace{0.83328pt}}}({\xi }) = ({\hspace{0.83328pt}\overline{\hspace{-0.83328pt}\Sigma \hspace{-0.83328pt}}\hspace{0.83328pt}}{\Sigma }^{-1})^{12} {\xi } ({\Sigma }^{-1}{\hspace{0.83328pt}\overline{\hspace{-0.83328pt}\Sigma \hspace{-0.83328pt}}\hspace{0.83328pt}})^{12}.\) 
